Tests for Convergence

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Math141 Farber

Summary of Convergence and Divergence Tests for Infinite Series

 

Test

Series

Convergence or divergence

Comments

nth term test for divergence

Diverges if

Use this test first if divergence is suspected. However, test is inconclusive if , so try another test.

Geometric series

(a) Converges if | r | < 1

 

(b) Diverges if

The first term is a and each term is a multiple, r of the previous term. Also useful for comparison tests if the nth term of a series is similar in magnitude to r n-1. If |r|< 1,   .

p-series

(a) Converges if p > 1

(b) Diverges if

Useful for comparison tests if the nth term of a series is similar in magnitude to .

Integral Test

(a) Converges if converges

(b) Diverges if diverges

The function É obtained from must be continuous, positive, decreasing. Use if you can integrate the function.

Individual Comparison

Test

Given

compare to known

to converge or to diverge

(a) If converges and for every n then converges

(b) If diverges and for every n then diverges

The comparison series is often a geometric series or a p-series.

To show convergence, you must find a series known to converge that is greater than the given series.

To show divergence, you must find a series known to diverge that is smaller than the given series.

Hints: ln n < n and |sin n | and |cos n| are always less than or equal to 1.

Limit Comparison

Test

,

(a) If for some positive real number L then both series converge or both diverge.

(b) If the limit is 0 and bn is known to converge, then an converges too.

(c) If the limit is and bn is known to diverge, an diverges too.

To find bn in the limit comparison test, consider only the terms in an that have the greatest effect on the magnitude. Use LíHopitalís rule when needed. This often is helpful when ln n appears.

Ratio

Test

If , the series

(a) converges (absolutely) if L <1.

(b) diverges if L >1 (or )

Inconclusive if L = 1. Useful if an involves factorials or nth powers. If all terms are positive then absolute value sign may be disregarded.

(n+1)!/n! = n+1

2n+1/2n = 2

n +1 term for (2n)! is (2(n+1))! = (2n + 2)!

Root Test

If , the series

(a) converges (absolutely) if L <1.

(b) diverges if L >1 (or )

Inconclusive if L = 1. Useful if nth term involves nth powers.

Alternating series

Test

Converges if

(a)

(b) for every n >M

(c)

Applies only to alternating series. Series diverges if any one of the conditions is not met. Exponent on numeric factor may be n or n+1 depending on whether the terms are negative for n odd or even.

If converges then converges

If a series converges absolutely, then it converges. A series that converges but does not converge absolutely is called conditionally convergent.

Useful for series containing both positive and negative terms that do not alternate. Use for series with trig functions. This is also useful when you know the non-negative series converges. Adding some negative terms will only make the series converge more quickly. Note the converse to this is not true. For example, the alternating harmonic series converges by the alternating series test but the nonnegative harmonic series diverges.

 

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